

In [1]:

    
# Graphing helper function
def setup_graph(title='', x_label='', y_label='', fig_size=None):
    fig = plt.figure()
    if fig_size != None:
        fig.set_size_inches(fig_size[0], fig_size[1])
    ax = fig.add_subplot(111)
    ax.set_title(title)
    ax.set_xlabel(x_label)
    ax.set_ylabel(y_label)

Wave Deconvolution - Multiplication and Addition

So a complex wave is made of simple waves.
How can you break a complex wave into it's simple component waves?

Answer: Multiplying a wave by another wave of frequency 100Hz cancels out all of the other component waves and leaves only the 100Hz component (kind of).



In [27]:

    
t = linspace(0, 3, 200)
freq_1hz_amp_10 = 10 * sin(1 * 2*pi*t)
freq_3hz_amp_5 =   5 * sin(3 * 2*pi*t)
complex_wave = freq_1hz_amp_10 + freq_3hz_amp_5

setup_graph(x_label='time (in seconds)', y_label='amplitude', title='original wave', fig_size=(12,6))
_ = plot(t, complex_wave)

Multiply complex wave by 1Hz wave



In [28]:

    
freq_1hz = sin(1 * 2*pi*t)
setup_graph(x_label='time (in seconds)', y_label='amplitude', title='original wave * 1Hz wave', fig_size=(12,6))
_ = plot(t, complex_wave * freq_1hz)



In [29]:

    
sum(complex_wave*freq_1hz)









    Out[29]:





994.99999999999977



In [30]:

    
print "Amplitude of 1hz component: ", sum(complex_wave*freq_1hz) * 2.0 * 1.0/len(complex_wave)









    



Amplitude of 1hz component:  9.95

Multiply complex wave by 3Hz wave

Notice that more of the graph is above the x-axis then below it.



In [31]:

    
freq_3hz = sin(3 * 2*pi*t)
setup_graph(x_label='time (in seconds)', y_label='amplitude', title='complex wave * 3Hz wave', fig_size=(12,6))
_ = plot(t, complex_wave * freq_3hz)



In [32]:

    
sum(complex_wave*freq_3hz)









    Out[32]:





497.5



In [33]:

    
print "Amplitude of 3hz component: ", sum(complex_wave*freq_3hz) * 2.0/len(complex_wave)









    



Amplitude of 3hz component:  4.975

Multiply complex wave by 2Hz wave

Notice that an equal amount of the graph is above the x-axis as below it.



In [34]:

    
freq_2hz = sin(2 * 2*pi*t) 
setup_graph(x_label='time (in seconds)', y_label='amplitude', title='complex wave * 2Hz wave', fig_size=(12,6))
_ = plot(t, complex_wave * freq_2hz)



In [35]:

    
sum(complex_wave*freq_2hz)









    Out[35]:





1.4549472737712679e-13



In [36]:

    
# Very close to 0
print "Amplitude of 3hz component: ", sum(complex_wave*freq_2hz) * 2.0/len(complex_wave)









    



Amplitude of 3hz component:  1.45494727377e-15



In [37]:

    
# Same with 4Hz - close to 0
freq_4hz = sin(4 * 2*pi*t) 
sum(complex_wave*freq_4hz)









    Out[37]:





-5.0848214527832113e-14

So how does this work?

The summation of complex wave multiplied by simple wave of a given frequency leaves us with the "power" of that simple wave.